Scaling law and reduced models for epitaxially strained crystalline - PowerPoint PPT Presentation

Scaling law and reduced models for epitaxially strained crystalline films Michael Goldman MPI, Leipzig Joint work with B. Zwicknagl

Introduction The Mathematical Model Scaling Law Reduced Models

Introduction Epitaxially strained crystalline films are obtained by deposing thin layers on a thick substrate Deposed atoms Substrate Example : In-GaAs/GaAs or SiGe/Si.

Governing mechanism There is a mismatch between the lattice parameters of the two crystals The deposit layer is strained and the atoms try to rearrange for releasing elastic energy but this migration is also energetically expensive ⇒ interaction between bulk and surface energy. =

Numerical and experimental observations Existence of several regimes depending on the volume of the thin layer and of the mismatch • For small volumes, the flat configuration is favored • Above a certain threshold, the flat configuration is not stable anymore and the film develops corrugations • For higher values of the volume/mismatch, there is formation of isolated islands Goal: Understand these different regimes.

Surface roughening in SiGe/Si, images from Gao, Nix, Surface roughening of heteroepitaxial thin films , 1999.

Formation of islands, images from Gray, Hull and Floro Formation of one-dimensional surface grooves from pit instabilities in annealed SiGe/Si(100) epitaxial films , 2004.

Experimental results Example of cusps, images from Chen, Jesson, Pennycook, Thundat, and Warmack, Cuspidal pit formation during the growth of SixGe1-x strained films , 1995

Numerical simulations Numerical simulations from Bonnetier and Chambolle, Computing the Equilibrium Configuration of Epitaxially Strained Crystalline Films , 2002. See also the numerical simulations of University of Cambridge, DoITPoMS, http://www.doitpoms.ac.uk/tlplib/epitaxial-growth/index.php

Applications These epitaxially grown thin films are used for • Optical and optoelectric devices (quantum dot laser). • Semiconductors. • Information storage. • Nanotechnology.

The Mathematical Model The film is taken to be the subgraph Ω h of a function h : [0 , 1] → R + h ( x ) Ω h u ( x , y ) = e 0 ( x , y ) The substrate is considered as rigid hence in the substrate, the deformation is equal to e 0 ( x , y ) where e 0 is the mismatch.

The energy Let W : R 4 → R + be the stored elastic energy then we consider the variational problem: � � 1 � 1 + | h ′ | 2 F d , e 0 ( u , h ) := W ( ∇ u ) + Ω h 0 under the conditions that � 1 u ( x , 0) = e 0 ( x , 0) and h = d 0 Remark: most of the works consider energies W depending only on the symmetric part of the gradient.

Contributions of each term in the energy • Due to the mismatch, there are no stress free configurations. • In order to release elastic energy, the bulk term favors creation of singularities. • On the other hand, the surface term tends to avoid too many oscillations.

Regularity results (in the geometrically linear setting) cusp cut Theorem [Chambolle-Larsen 03, Fonseca-Fusco-Leoni-Morini 07] The profile h is regular out of a finite number of cuts and cuts. Moreover the film satisfies the zero angle condition.

Regularity results continued Theorem [Fusco-Morini 12] • For small mismatch, the flat configuration is minimzing (no matter how big is d ). • For greater mismatch, the following holds: 1. for d ≤ d 0 , the flat configuration is minimizing 2. for d ≤ d 1 the flat configuration is locally minimizing 3. for d ≤ d 2 , the flat configuration is not locally minimizing but every minimizer is smooth

Other results in the litterature • Physical and engineering: Spencer-Meiron 94, Spencer-Tersoff 10, Gao-Nix 99. • Regularity, relaxation and approximation: Bonnetier-Chambolle 02, Chambolle-Larsen 03, Fonseca-Fusco-Leoni-Morini 07, Chambolle-Solci 07, Fusco-Morini 12. • Time evolution: Fonseca-Fusco-Leoni-Morini 12, Piovano 12.

Other results in the litterature • Physical and engineering: Spencer-Meiron 94, Spencer-Tersoff 10, Gao-Nix 99. • Regularity, relaxation and approximation: Bonnetier-Chambolle 02, Chambolle-Larsen 03, Fonseca-Fusco-Leoni-Morini 07, Chambolle-Solci 07, Fusco-Morini 12. • Time evolution: Fonseca-Fusco-Leoni-Morini 12, Piovano 12. No rigourous result on the formation of the islands!

The main result We will assume that Hypothesis (H1) W ≥ 0 (H2) there exists C > 0 and p > 1 such that C ( | A | p + 1) ≥ W ( A ) ≥ 1 C ( | A | p − 1) ∀ A ∈ R 2 × 2 . Theorem Under these assumptions, for every e 0 > 0 and d > 0 there holds u , h F e 0 , d ( u , h ) ≃ max(1 , d , e p / 3 d 2 / 3 ) . min 0 Remark: • Thanks to (H2), it is enough considering W ( ∇ u ) = |∇ u | p . • Works also in the geometrically linear setting.

Heuristic explanation of the scaling We consider for simplicity here p = 2 so that � � 1 � |∇ u | 2 + 1 + | h ′ | 2 F e 0 , d ( u , h ) = Ω h 0 If Ω h ∩ { y = 0 } = [ a , a + ℓ ] then since | Ω h | = d , � 1 � 1 + | h ′ | 2 ≥ d ℓ . 0 On the other hand � |∇ u | 2 ≃ e 2 0 | u | 2 min H 1 / 2 ( a , a + ℓ ) ℓ u ( x , 0)= e 0 ( x , 0) Ω h ≃ e 2 0 ℓ 2

Putting these together we find that 0 ℓ 2 + d F e 0 , d ( u , h ) � e 2 ℓ � � 1 / 3 d Optimizing in ℓ , we find that ℓ min ≃ min(1 , ). So that two e 2 0 regimes appear: � � 1 / 3 � � 1 / 3 d d • If ≤ 1, we have ℓ min = and e 2 e 2 0 0 min F e 0 , d ≃ e 2 / 3 d 2 / 3 . 0 � � 1 / 3 d • If ≥ 1, the flat configuartion is favored and e 2 0 min F e 0 , d ≃ e 2 0 + d ≃ d . Difficulty: when h ( x ) ≪ 1, the constant in the trace inequality degenerate i.e. � |∇ u | 2 � e 2 0 | u | 2 min H 1 / 2 ( a , a + ℓ ) u ( x , 0)= e 0 ( x , 0) Ω h

The Strategy To prove this kind of scaling laws, the general strategy is • To get the upper bound by construction. • To prove an ansatz free lower bound. In many related results (see Kohn-M¨ uller, Choksi-Conti-Kohn-Otto, Bella-Kohn, Capella-Otto...) the lower bound is obtained via an interpolation inequality. Here it will not be the case.

Preamble, playing with rectangles For u ∈ W 1 , p ([0 , ℓ ] × [0 , L ]) let u ( x , y ) = 1 ℓ u ( ℓ x , ℓ y ) ∈ W 1 , p ([0 , 1] × [0 , L /ℓ ]) . ˜ L Then ∇ ˜ u ( x , y ) = ∇ u ( x , y ) and � � |∇ u | p = ℓ 2 u | p |∇ ˜ ℓ [0 ,ℓ ] × [0 , L ] [0 , 1] × [0 , L /ℓ ] Fondamental lemma � � |∇ u | p = e p 0 ℓ 2 |∇ u | p min min u ( x , 0)= e 0 ( x , 0) u ( x , 0)=( x , 0) [0 ,ℓ ] × [0 , L ] [0 , 1] × [0 , L /ℓ ]

The upper bound By the considerations above, for the upper bound, it is enough � � 1 / 3 � � d considering a rectangle [0 , ℓ ] × [0 , d /ℓ ] with ℓ = min 1 , e p 0 and � � � 1 − 1 ( e 0 x ℓ y , 0) if 0 ≤ y ≤ ℓ, u ( x , y ) = 0 else . ℓ ≃ max(1 , d , e p / 3 Then F e 0 , d ( u , h ) ≃ ℓ 2 e p 0 + 1 + d d 2 / 3 ) . 0

The lower bound: setting the notations Since F d , e 0 ( u , h ) ≥ 1 + d , we can assume e p / 3 d 2 / 3 ≥ max(1 , d ). 0 Let : d y 0 := d 2 / 3 . � e p / 3 2 0 ℓ := H 1 (Ω h ∩ ( I × { y 0 } )). d 2 d 1 I ℓ := Ω h ∩ ( I × { y 0 } ). b 2 a 2 a 1 b 1 I ℓ = ∪ n i =1 [ a i , b i ]. ℓ 1 ℓ 2 y 0 ℓ i := b i − a i . d i := | Ω h ∩ ([ a i , b i ] × [ y 0 , + ∞ )) | . Then � n i =1 ℓ i = ℓ and � n i =1 d i ≥ d − y 0 ≥ Cd .

� � 1 / 3 d First possibility: ℓ i ≤ for all i = 1 , . . . , n e p 0 In this case, the surface energy is sufficient to get � 1 � 1 + | h ′ | 2 dx F d , e 0 ( u , h ) ≥ 0 � b i � n � 1 + | h ′ | 2 dx ≥ a i i =1 � e p � 1 / 3 n n � � d i 0 ≥ ≥ d i ℓ i d i =1 i =1 d 2 / 3 . ≥ Ce p / 3 0 And we are done.

� � 1 / 3 d Second possibility, ℓ 1 ≥ e p 0 In this case, we focus on the elastic energy and find � |∇ u | p dxdy F d , e 0 ( u , h ) ≥ [ a 1 , b 1 ] × [0 , y 0 ] � ≥ ℓ 2 1 e p |∇ u | p dxdy . min 0 v ( x , 0)=( x , 0) [0 , 1] × [0 , y 0 /ℓ 1 ] Problem: It can happen that y 0 /ℓ 1 ≪ 1...

� � 1 / 3 d Second possibility, ℓ 1 ≥ e p 0 In this case, we focus on the elastic energy and find � |∇ u | p dxdy F d , e 0 ( u , h ) ≥ [ a 1 , b 1 ] × [0 , y 0 ] � ≥ ℓ 2 1 e p |∇ u | p dxdy . min 0 v ( x , 0)=( x , 0) [0 , 1] × [0 , y 0 /ℓ 1 ] Problem: It can happen that y 0 /ℓ 1 ≪ 1... = ⇒ We have to control how � |∇ u | p dxdy → 0 min v ( x , 0)=( x , 0) [0 , 1] × [0 ,ε ] when ε → 0.

Theorem (Dimension reduction) There holds � 1 |∇ u | p dxdy ≥ 1 . lim min ε ε → 0 + u ( x , 0)=( x , 0) [0 , 1] × [0 ,ε ] This is a simplified version of the Le Dret-Raoult proof of dimension reduction. Remark: For p = 2, using Fourier methods, it can be seen that � + ∞ � 1 |∇ u | 2 dxdy ≃ min | k | 3 (1 − exp( − 2 π k ε )) u ( x , 0)=( x , 0) [0 , 1] × [0 ,ε ] k =1 + ∞ � 1 ≥ 2 πε | k | 3 k =1